My read is that once the president is elected, there is no longer an eligibility issue because he is already the president.
That is a crazy statement. If some illegal alien can trick his way by fraud, deceit, bribery, blackmail and forged documents into the presidency, and is found out, he’s in like flynn? Ollie ollie in free? Ha ha?
That’s really a crazy statement.
Elected? No.
Certified by a special Joint Session of Congress convened to hear objections? Perhaps.
My take is that there is no recourse for objections which existed at the time of the Special Joint Session, but were not placed on the floor.
In this case (the Hawaii documents), I think that the issue was well known on January 6, 2009, and the threshold of one Senator and one Member of Congress objecting was not met.
It is virtually impossible that a court will substitute its judgement for that of Congress in this matter, which appears to be under their sole purview by Article II and Amendment XII.
If a new issue arose during a Presidential term (i.e., one which Congress could not have considered at the time the electoral vote was counted), it is possible that a Federal court could make a ruling on that new issue.